2, 4, 8, 16, 32...is a geometric sequence in which each number is obtained by multiplying 2 to the previous term.
The terms geometric sequence and progression mean the same thing.
Common ratioCommon ratio is the fixed quantity that is multiplied to any term in a geometric sequence get the next term. For example, in the above geometric sequence, the common ratio is 2. Common ratio is represented by the a small or capital 'r'.
To calculate the common ratio of a given geometric sequence, divide the second term by the first term. For example, in the following geometric sequence:
1, 3, 9, 27, 81, 243, ..common ratio is 3 because second term, 3, divided by first term, 1, equals 3.
Infinite geometric progressionAn infinite geometric progression/sequence is one which has no last term. It's total number of terms is not known and so, by multiplying the common ratio to the previous term, it keeps going on. It is denoted by a ... after writing a few first terms. For example, the above geometric progression 1, 3, 9, 27, 81, 243, ... has a ... after it denoting that it is an infinite geometric progression.
Geometric seriesA geometric series is a geometric sequence or progression which is written with + signs separating its terms instead of commas. For example the above geometric progression can be written as a geometric series in the following way:
1 + 3 + 9 + 29 + 81 + 243 + ...The formulas for a geometric series are same as those for a geometric progression.
General termThe nth term of a geometric sequence having first term 'a' and common ratio 'r' is given by:
- When r ≠ 1, | r | < 1
- When r ≠ 1, r < -1 or r > 1
- When r = 1
Sn = na